#1**+2 **

Let the side of the octagon = S

And one of the yellow regions is a trapezoid

And the height of one of these trapezoids = S / √2

And the base of one of these trapezoids = S + 2S/√2 = (1 +√2)S

So....the area of one of the trapezoids = (1/2) (S/√2) ( S + (1 + √2) S) =

(√2/4) ( 2 + √2)S^2 =

(1/4) (2 + 2√2)S^2 =

(1/2) ( 1 + √2)S^2

So the two trapezoids have an area of ( 1 + √2)S^2 = the yellow area

And the area of the blue rectangle = S * Base of the trapezoid =

S * (1 + √2)S = (1 + √2)S^2

So.....the yellow and blue areas are equal !!!!

CPhill Dec 21, 2019